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Mathematics > Representation Theory
Title: Arithmetic geometry of character varieties with regular monodromy, I
(Submitted on 6 Sep 2022 (v1), last revised 15 Sep 2023 (this version, v4))
Abstract: We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.
Submission history
From: Masoud Kamgarpour [view email][v1] Tue, 6 Sep 2022 00:41:01 GMT (22kb)
[v2] Thu, 16 Feb 2023 07:26:23 GMT (27kb)
[v3] Thu, 22 Jun 2023 18:09:16 GMT (30kb)
[v4] Fri, 15 Sep 2023 01:25:15 GMT (31kb)
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